Non-Kähler geometry and pseudoconcave structures

The article delves into the tension between local smoothness and global rigor in complex geometry, focusing on the works of Naohiko Kasuya. The author contrasts harmonious Kähler manifolds, where analysis and topology are consistent, with less regular areas: non-Kähler and strongly pseudoconcave surfaces. The text explains that while for compact surfaces Kählerianity can be determined by the parity of the first Betti number, in open manifolds the situation becomes significantly more complex. The analysis points to deep connections between these structures and contact and symplectic geometry, shifting the center of gravity from classical algebraic models toward peripheral, more elusive forms of complex geometry.